Many machine learning methods such as classification via support vector machines or regression [1]}, statistical tests [2]}, [3]}, or generative learning [4]}, [5]} can be adapted to the setting of reproducing kernel Hilbert spaces.
Through the design of adapted kernels, this enables the application of these methods to non-Euclidean data such as strings, sequences and graphs [6]}.
A typical application is the modeling and classification of proteins [7]} using graph kernels [8]}.
The MMDs leverage these existing kernel constructions to compare collections of points (point clouds) or more generally distributions representing data [2]}, [3]}.
In practice, one usually considers discrete empirical distributions, \((_N,_N)\) , often assumed to be drawn from some unknown measures \((,)\) .
This is at the heart of two-sample tests to check whether \(=\beta \)  by checking if \({_N - _N}_{^*}\)  is small enough.
Another application of kernels is to perform density estimation [11]} by smoothing a discrete measure \(_N\)  to obtain \((k\star _N)(x) \triangleq \int k(x,y)_N(y)\) .
When \(_N=\sum _i _i_{x_i}\) , this smoothing reads \((k\star _N)(x) = \sum _i _i k(x,x_i)\) .
